1. TOPIC 0-FUNDAMENTAL CONCEPTS OF ALGEBRA (MAT0114)
REAL NUMBERS SYSTEM
EXPONENTS: LAW OF INDICES
ALGEBRAIC EXPRESSION
RADICALS

2. 1. REAL NUMBER SYSTEM
Real
Rational
Integer
Negative
Zero
Positive
Natural
Non integer
Terminate Decimal
Decimal repeat in cycle
Irrational
Decimal does not repeat in cycle
Example1: Classify the numbers into their type (Natural, Integer, Rational, Real).
2.33, −1.5, 7 4 , 𝜋, 6 , − 11 2 , 2

3. INTERVALS Open interval : , Close interval: ,
Union: take all the values in the interval.
−2,3 ∪ 2,10 =
Intersection: take the intersection values only.
3 ,12)∩ −2,8 =

4. 2. EXPONENTS & RATIONAL EXPONENT
LAW OF EXPONENT
LAW OF RATIONAL EXPONENT
1) 𝑎 𝑚 × 𝑎 𝑛 = 𝑎 𝑚+𝑛
2) 𝑎 𝑚 ÷ 𝑎 𝑛 = 𝑎 𝑚−𝑛
3) 𝑎 𝑚 𝑛 = 𝑎 𝑚𝑛
4) 𝑎𝑏 𝑛 = 𝑎 𝑛 𝑏 𝑛
5) 𝑎 𝑏 𝑛 = 𝑎 𝑛 𝑏 𝑛
1) 𝑎 1 𝑛 = 𝑛 𝑎
2) 𝑎 𝑚 𝑛 = 𝑛 𝑎 𝑚 = 𝑛 𝑎 𝑚
3) 𝑎 𝑚 𝑛 = 𝑎 1 𝑛 𝑚 = 𝑎 𝑚 1 𝑛
THEOREM ON NEGATIVE EXPONENTS
1) 𝑎 −𝑛 = 1 𝑎 𝑛 ) 𝑎 −𝑚 𝑏 −𝑛 = 𝑏 𝑛 𝑎 𝑚 ) 𝑎 𝑏 −𝑛 = 𝑏 𝑎 𝑛

5. 3. Algebraic expression
Algebraic expression is obtained by applying additions, subtractions, multiplications, divisions, powers or taking roots to collection of variables and real numbers.
Simplify the algebraic expression:
a) 2𝑢+3 𝑢−4 +4𝑢 𝑢−2 =

6. PRODUCT FORMULAS 1) 𝑥+𝑦 𝑥−𝑦 = 𝑥 2 − 𝑦 2 2) 𝑥+𝑦 2 = 𝑥 2 +2𝑥𝑦+ 𝑦 2
1) 𝑥+𝑦 𝑥−𝑦 = 𝑥 2 − 𝑦 2
2) 𝑥+𝑦 2 = 𝑥 2 +2𝑥𝑦+ 𝑦 2
3) 𝑥−𝑦 2 = 𝑥 2 −2𝑥𝑦+ 𝑦 2
4) 𝑥+𝑦 3 = 𝑥 3 +3 𝑥 2 𝑦+3𝑥 𝑦 2 + 𝑦 3
5) 𝑥−𝑦 3 = 𝑥 3 −3 𝑥 2 𝑦+3𝑥 𝑦 2 − 𝑦 3
EXERCISES: Find the product
i) 2 𝑟 2 − 𝑠 2 𝑟 2 + 𝑠 =
ii) 2𝑥+3𝑦 3 =

7. FACTORING FORMULAS 1) Difference of two squares: 𝑥 2 − 𝑦 2 = 𝑥+𝑦 𝑥−𝑦
2) Difference of two cubes: 𝑥 3 − 𝑦 3 = 𝑥−𝑦 𝑥 2 +𝑥𝑦+ 𝑦 2
3) Sum of two cubes: 𝑥 3 + 𝑦 3 = 𝑥+𝑦 𝑥 2 −𝑥𝑦+ 𝑦 2
EXERCISES: Factor each polynomials
i) 36 𝑟 2 −25 𝑡 2 =
ii) 125 𝑥 3 −8=

8. FACTORING BY TRIAL & ERROR
Trying various possibilities to factor the algebraic expression.
Factorize:
i) 6 𝑥 2 −7𝑥−3=

9. FACTORING BY GROUPING
If a sum contains four or more terms, it may be possible to group the terms in a suitable manner then find a factorization by using distributive properties.
Factor each expression:
i) 3 𝑥 3 +3 𝑥 2 −27𝑥−27=

10. ABSOLUTE VALUE
- is defined as the distance from the origin to the specific point/value on real number line.
i) if 𝑎≥0, 𝑡ℎ𝑒𝑛 𝑎 =𝑎
ii) if 𝑎<0, 𝑡ℎ𝑒𝑛 𝑎 =−(𝑎)
Example 1: Evaluate
a) 7−12 =
b) 𝜋−6 =

11. 4.0 RADICALS PROPERTIES OF nth ROOT 1) 𝑛 𝑎𝑏 = 𝑛 𝑎 𝑛 𝑏
1) 𝑛 𝑎𝑏 = 𝑛 𝑎 𝑛 𝑏
2) 𝑛 𝑎 𝑏 = 𝑛 𝑎 𝑛 𝑏
3) 𝑚 𝑛 𝑎 = 𝑚𝑛 𝑎
4) 𝑛 𝑎 𝑛 = 𝑎 if n is even
5) 𝑛 𝑎 𝑛 =𝑎 if n is odd

12. RATIONALIZE DENOMINATOR OF QUOTIENTS
FACTOR IN DENOMINATOR
MULTIPLY NUMERATOR AND DENOMINATOR BY:
RESULTING FACTOR: 𝑎
𝑎 2 =𝑎
3 𝑎
3 𝑎 2
3 𝑎 3 =𝑎
7 𝑎 3
7 𝑎 4
7 𝑎 7 =𝑎
Examples: Simplify and rationalize the denominator a)
b) 2 𝑥
c) 𝑥 4 𝑦 4 9𝑥